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Theorem List for Metamath Proof Explorer - 28801-28900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorembrelg 28801 Two things in a binary relation belong to the relation's domain. (Contributed by Thierry Arnoux, 29-Aug-2017.)
((𝑅 ⊆ (𝐶 × 𝐷) ∧ 𝐴𝑅𝐵) → (𝐴𝐶𝐵𝐷))

Theorembr8d 28802* Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by Thierry Arnoux, 21-Mar-2019.)
(𝑎 = 𝐴 → (𝜓𝜒))    &   (𝑏 = 𝐵 → (𝜒𝜃))    &   (𝑐 = 𝐶 → (𝜃𝜏))    &   (𝑑 = 𝐷 → (𝜏𝜂))    &   (𝑒 = 𝐸 → (𝜂𝜁))    &   (𝑓 = 𝐹 → (𝜁𝜎))    &   (𝑔 = 𝐺 → (𝜎𝜌))    &   ( = 𝐻 → (𝜌𝜇))    &   (𝜑𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑎𝑃𝑏𝑃𝑐𝑃𝑑𝑃𝑒𝑃𝑓𝑃𝑔𝑃𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ⟩⟩ ∧ 𝜓)})    &   (𝜑𝐴𝑃)    &   (𝜑𝐵𝑃)    &   (𝜑𝐶𝑃)    &   (𝜑𝐷𝑃)    &   (𝜑𝐸𝑃)    &   (𝜑𝐹𝑃)    &   (𝜑𝐺𝑃)    &   (𝜑𝐻𝑃)       (𝜑 → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜇))

Theoremopabdm 28803* Domain of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → dom 𝑅 = {𝑥 ∣ ∃𝑦𝜑})

Theoremopabrn 28804* Range of an ordered-pair class abstraction. (Contributed by Thierry Arnoux, 31-Aug-2017.)
(𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ran 𝑅 = {𝑦 ∣ ∃𝑥𝜑})

Theoremssrelf 28805* A subclass relationship depends only on a relation's ordered pairs. Theorem 3.2(i) of [Monk1] p. 33. (Contributed by NM, 2-Aug-1994.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Revised by Thierry Arnoux, 6-Nov-2017.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝐴    &   𝑦𝐴    &   𝑥𝐵    &   𝑦𝐵       (Rel 𝐴 → (𝐴𝐵 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 → ⟨𝑥, 𝑦⟩ ∈ 𝐵)))

Theoremeqrelrd2 28806* A version of eqrelrdv2 5142 with explicit non-free declarations. (Contributed by Thierry Arnoux, 28-Aug-2017.)
𝑥𝜑    &   𝑦𝜑    &   𝑥𝐴    &   𝑦𝐴    &   𝑥𝐵    &   𝑦𝐵    &   (𝜑 → (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐵))       (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵)

Theoremerbr3b 28807 Biconditional for equivalent elements. (Contributed by Thierry Arnoux, 6-Jan-2020.)
((𝑅 Er 𝑋𝐴𝑅𝐵) → (𝐴𝑅𝐶𝐵𝑅𝐶))

Theoremiunsnima 28808 Image of a singleton by an indexed union involving that singleton. (Contributed by Thierry Arnoux, 10-Apr-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑥𝐴) → 𝐵𝑊)       ((𝜑𝑥𝐴) → ( 𝑥𝐴 ({𝑥} × 𝐵) “ {𝑥}) = 𝐵)

Theoremmptexgf 28809 If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) (Revised by Thierry Arnoux, 17-May-2020.)
𝑥𝐴       (𝐴𝑉 → (𝑥𝐴𝐵) ∈ V)

Theoremac6sf2 28810* Alternate version of ac6 9185 with bound-variable hypothesis. (Contributed by NM, 2-Mar-2008.) (Revised by Thierry Arnoux, 17-May-2020.)
𝑦𝐵    &   𝑦𝜓    &   𝐴 ∈ V    &   (𝑦 = (𝑓𝑥) → (𝜑𝜓))       (∀𝑥𝐴𝑦𝐵 𝜑 → ∃𝑓(𝑓:𝐴𝐵 ∧ ∀𝑥𝐴 𝜓))

Theoremidssxp 28811 A diagonal set as a subset of a Cartesian product. (Contributed by Thierry Arnoux, 29-Dec-2019.)
( I ↾ 𝐴) ⊆ (𝐴 × 𝐴)

Theoremfnresin 28812 Restriction of a function with a subclass of its domain. (Contributed by Thierry Arnoux, 10-Oct-2017.)
(𝐹 Fn 𝐴 → (𝐹𝐵) Fn (𝐴𝐵))

Theoremf1o3d 28813* Describe an implicit one-to-one onto function. (Contributed by Thierry Arnoux, 23-Apr-2017.)
(𝜑𝐹 = (𝑥𝐴𝐶))    &   ((𝜑𝑥𝐴) → 𝐶𝐵)    &   ((𝜑𝑦𝐵) → 𝐷𝐴)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → (𝑥 = 𝐷𝑦 = 𝐶))       (𝜑 → (𝐹:𝐴1-1-onto𝐵𝐹 = (𝑦𝐵𝐷)))

Theoremrinvf1o 28814 Sufficient conditions for the restriction of an involution to be a bijection. (Contributed by Thierry Arnoux, 7-Dec-2016.)
Fun 𝐹    &   𝐹 = 𝐹    &   (𝐹𝐴) ⊆ 𝐵    &   (𝐹𝐵) ⊆ 𝐴    &   𝐴 ⊆ dom 𝐹    &   𝐵 ⊆ dom 𝐹       (𝐹𝐴):𝐴1-1-onto𝐵

Theoremfresf1o 28815 Conditions for a restriction to be a one-to-one onto function. (Contributed by Thierry Arnoux, 7-Dec-2016.)
((Fun 𝐹𝐶 ⊆ ran 𝐹 ∧ Fun (𝐹𝐶)) → (𝐹 ↾ (𝐹𝐶)):(𝐹𝐶)–1-1-onto𝐶)

Theoremf1mptrn 28816* Express injection for a mapping operation. (Contributed by Thierry Arnoux, 3-May-2020.)
((𝜑𝑥𝐴) → 𝐵𝐶)    &   ((𝜑𝑦𝐶) → ∃!𝑥𝐴 𝑦 = 𝐵)       (𝜑 → Fun (𝑥𝐴𝐵))

Theoremdfimafnf 28817* Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.) (Revised by Thierry Arnoux, 24-Apr-2017.)
𝑥𝐴    &   𝑥𝐹       ((Fun 𝐹𝐴 ⊆ dom 𝐹) → (𝐹𝐴) = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹𝑥)})

Theoremfunimass4f 28818 Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.)
𝑥𝐴    &   𝑥𝐵    &   𝑥𝐹       ((Fun 𝐹𝐴 ⊆ dom 𝐹) → ((𝐹𝐴) ⊆ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))

Theoremsuppss2f 28819* Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.) (Revised by AV, 1-Sep-2020.)
𝑘𝜑    &   𝑘𝐴    &   𝑘𝑊    &   ((𝜑𝑘 ∈ (𝐴𝑊)) → 𝐵 = 𝑍)    &   (𝜑𝐴𝑉)       (𝜑 → ((𝑘𝐴𝐵) supp 𝑍) ⊆ 𝑊)

Theoremfovcld 28820 Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.)
(𝜑𝐹:(𝑅 × 𝑆)⟶𝐶)       ((𝜑𝐴𝑅𝐵𝑆) → (𝐴𝐹𝐵) ∈ 𝐶)

Theoremofrn 28821 The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑+ :(𝐵 × 𝐵)⟶𝐶)    &   (𝜑𝐴𝑉)       (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ 𝐶)

Theoremofrn2 28822 The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑+ :(𝐵 × 𝐵)⟶𝐶)    &   (𝜑𝐴𝑉)       (𝜑 → ran (𝐹𝑓 + 𝐺) ⊆ ( + “ (ran 𝐹 × ran 𝐺)))

Theoremoff2 28823* The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.)
((𝜑 ∧ (𝑥𝑆𝑦𝑇)) → (𝑥𝑅𝑦) ∈ 𝑈)    &   (𝜑𝐹:𝐴𝑆)    &   (𝜑𝐺:𝐵𝑇)    &   (𝜑𝐴𝑉)    &   (𝜑𝐵𝑊)    &   (𝜑 → (𝐴𝐵) = 𝐶)       (𝜑 → (𝐹𝑓 𝑅𝐺):𝐶𝑈)

Theoremofresid 28824 Applying an operation restricted to the range of the functions does not change the function operation. (Contributed by Thierry Arnoux, 14-Feb-2018.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐵)    &   (𝜑𝐴𝑉)       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝐹𝑓 (𝑅 ↾ (𝐵 × 𝐵))𝐺))

Theoremfimarab 28825* Expressing the image of a set as a restricted abstract builder. (Contributed by Thierry Arnoux, 27-Jan-2020.)
((𝐹:𝐴𝐵𝑋𝐴) → (𝐹𝑋) = {𝑦𝐵 ∣ ∃𝑥𝑋 (𝐹𝑥) = 𝑦})

Theoremunipreima 28826* Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)
(Fun 𝐹 → (𝐹 𝐴) = 𝑥𝐴 (𝐹𝑥))

Theoremsspreima 28827 The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)
((Fun 𝐹𝐴𝐵) → (𝐹𝐴) ⊆ (𝐹𝐵))

Theoremopfv 28828 Value of a function producing ordered pairs. (Contributed by Thierry Arnoux, 3-Jan-2017.)
(((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩)

Theoremxppreima 28829 The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 6-Jun-2017.)
((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (𝐹 “ (𝑌 × 𝑍)) = (((1st𝐹) “ 𝑌) ∩ ((2nd𝐹) “ 𝑍)))

Theoremxppreima2 28830* The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐶)    &   𝐻 = (𝑥𝐴 ↦ ⟨(𝐹𝑥), (𝐺𝑥)⟩)       (𝜑 → (𝐻 “ (𝑌 × 𝑍)) = ((𝐹𝑌) ∩ (𝐺𝑍)))

Theoremelunirn2 28831 Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.)
((Fun 𝐹𝐵 ∈ (𝐹𝐴)) → 𝐵 ran 𝐹)

Theoremabfmpunirn 28832* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})    &   {𝑦𝜑} ∈ V    &   (𝑦 = 𝐵 → (𝜑𝜓))       (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 𝜓))

Theoremrabfmpunirn 28833* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝑊𝜑})    &   𝑊 ∈ V    &   (𝑦 = 𝐵 → (𝜑𝜓))       (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 (𝐵𝑊𝜓))

Theoremabfmpeld 28834* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝜓})    &   (𝜑 → {𝑦𝜓} ∈ V)    &   (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜓𝜒)))       (𝜑 → ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜒)))

Theoremabfmpel 28835* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})    &   {𝑦𝜑} ∈ V    &   ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))       ((𝐴𝑉𝐵𝑊) → (𝐵 ∈ (𝐹𝐴) ↔ 𝜓))

TheoremfmptdF 28836 Domain and co-domain of the mapping operation; deduction form. This version of fmptd 6292 uses bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐶    &   ((𝜑𝑥𝐴) → 𝐵𝐶)    &   𝐹 = (𝑥𝐴𝐵)       (𝜑𝐹:𝐴𝐶)

Theoremmpteq12df 28837 An equality theorem for the maps to notation. (Contributed by Thierry Arnoux, 30-May-2020.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐶    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → (𝑥𝐴𝐵) = (𝑥𝐶𝐷))

Theoremresmptf 28838 Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)
𝑥𝐴    &   𝑥𝐵       (𝐵𝐴 → ((𝑥𝐴𝐶) ↾ 𝐵) = (𝑥𝐵𝐶))

Theoremfmptcof2 28839* Composition of two functions expressed as ordered-pair class abstractions. (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.) (Revised by Thierry Arnoux, 10-May-2017.)
𝑥𝑆    &   𝑦𝑇    &   𝑥𝐴    &   𝑥𝐵    &   𝑥𝜑    &   (𝜑 → ∀𝑥𝐴 𝑅𝐵)    &   (𝜑𝐹 = (𝑥𝐴𝑅))    &   (𝜑𝐺 = (𝑦𝐵𝑆))    &   (𝑦 = 𝑅𝑆 = 𝑇)       (𝜑 → (𝐺𝐹) = (𝑥𝐴𝑇))

Theoremfcomptf 28840* Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 6306. (Contributed by Thierry Arnoux, 30-Jun-2017.)
𝑥𝐵       ((𝐴:𝐷𝐸𝐵:𝐶𝐷) → (𝐴𝐵) = (𝑥𝐶 ↦ (𝐴‘(𝐵𝑥))))

Theoremacunirnmpt 28841* Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 6-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝐶 = ran (𝑗𝐴𝐵)       (𝜑 → ∃𝑓(𝑓:𝐶 𝐶 ∧ ∀𝑦𝐶𝑗𝐴 (𝑓𝑦) ∈ 𝐵))

Theoremacunirnmpt2 28842* Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝐶 = ran (𝑗𝐴𝐵)    &   (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)       (𝜑 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))

Theoremacunirnmpt2f 28843* Axiom of choice for the union of the range of a mapping to function. (Contributed by Thierry Arnoux, 7-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝑗𝐴    &   𝑗𝐶    &   𝑗𝐷    &   𝐶 = 𝑗𝐴 𝐵    &   (𝑗 = (𝑓𝑥) → 𝐵 = 𝐷)    &   ((𝜑𝑗𝐴) → 𝐵𝑊)       (𝜑 → ∃𝑓(𝑓:𝐶𝐴 ∧ ∀𝑥𝐶 𝑥𝐷))

Theoremaciunf1lem 28844* Choice in an index union. (Contributed by Thierry Arnoux, 8-Nov-2019.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵 ≠ ∅)    &   𝑗𝐴    &   ((𝜑𝑗𝐴) → 𝐵𝑊)       (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑥 𝑗𝐴 𝐵(2nd ‘(𝑓𝑥)) = 𝑥))

Theoremaciunf1 28845* Choice in an index union. (Contributed by Thierry Arnoux, 4-May-2020.)
(𝜑𝐴𝑉)    &   ((𝜑𝑗𝐴) → 𝐵𝑊)       (𝜑 → ∃𝑓(𝑓: 𝑗𝐴 𝐵1-1 𝑗𝐴 ({𝑗} × 𝐵) ∧ ∀𝑘 𝑗𝐴 𝐵(2nd ‘(𝑓𝑘)) = 𝑘))

Theoremcofmpt 28846* Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)
(𝜑𝐹:𝐶𝐷)    &   ((𝜑𝑥𝐴) → 𝐵𝐶)       (𝜑 → (𝐹 ∘ (𝑥𝐴𝐵)) = (𝑥𝐴 ↦ (𝐹𝐵)))

Theoremofoprabco 28847* Function operation as a composition with an operation. (Contributed by Thierry Arnoux, 4-Jun-2017.)
𝑎𝑀    &   (𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝑀 = (𝑎𝐴 ↦ ⟨(𝐹𝑎), (𝐺𝑎)⟩))    &   (𝜑𝑁 = (𝑥𝐵, 𝑦𝐶 ↦ (𝑥𝑅𝑦)))       (𝜑 → (𝐹𝑓 𝑅𝐺) = (𝑁𝑀))

Theoremofpreima 28848* Express the preimage of a function operation as a union of preimages. (Contributed by Thierry Arnoux, 8-Mar-2018.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 Fn (𝐵 × 𝐶))       (𝜑 → ((𝐹𝑓 𝑅𝐺) “ 𝐷) = 𝑝 ∈ (𝑅𝐷)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))

Theoremofpreima2 28849* Express the preimage of a function operation as a union of preimages. This version of ofpreima 28848 iterates the union over a smaller set. (Contributed by Thierry Arnoux, 8-Mar-2018.)
(𝜑𝐹:𝐴𝐵)    &   (𝜑𝐺:𝐴𝐶)    &   (𝜑𝐴𝑉)    &   (𝜑𝑅 Fn (𝐵 × 𝐶))       (𝜑 → ((𝐹𝑓 𝑅𝐺) “ 𝐷) = 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))

TheoremfuncnvmptOLD 28850* Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐹    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥(𝑥𝐴𝑦 = 𝐵)))

Theoremfuncnvmpt 28851* Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐹    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (Fun 𝐹 ↔ ∀𝑦∃*𝑥𝐴 𝑦 = 𝐵))

Theoremfuncnv5mpt 28852* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐹    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)    &   (𝑥 = 𝑧𝐵 = 𝐶)       (𝜑 → (Fun 𝐹 ↔ ∀𝑥𝐴𝑧𝐴 (𝑥 = 𝑧𝐵𝐶)))

Theoremfuncnv4mpt 28853* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐹    &   𝐹 = (𝑥𝐴𝐵)    &   ((𝜑𝑥𝐴) → 𝐵𝑉)       (𝜑 → (Fun 𝐹 ↔ ∀𝑖𝐴𝑗𝐴 (𝑖 = 𝑗𝑖 / 𝑥𝐵𝑗 / 𝑥𝐵)))

Theoremfgreu 28854* Exactly one point of a function's graph has a given first element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
((Fun 𝐹𝑋 ∈ dom 𝐹) → ∃!𝑝𝐹 𝑋 = (1st𝑝))

Theoremfcnvgreu 28855* If the converse of a relation 𝐴 is a function, exactly one point of its graph has a given second element (that is, function value). (Contributed by Thierry Arnoux, 1-Apr-2018.)
(((Rel 𝐴 ∧ Fun 𝐴) ∧ 𝑌 ∈ ran 𝐴) → ∃!𝑝𝐴 𝑌 = (2nd𝑝))

Theoremrnmpt2ss 28856* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 23-May-2017.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)       (∀𝑥𝐴𝑦𝐵 𝐶𝐷 → ran 𝐹𝐷)

TheoremmptssALT 28857* Deduce subset relation of mapping-to function graphs from a subset relation of domains. Alternative proof of mptss 5373. (Contributed by Thierry Arnoux, 30-May-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐵 → (𝑥𝐴𝐶) ⊆ (𝑥𝐵𝐶))

Theorempartfun 28858 Rewrite a function defined by parts, using a mapping and an if construct, into a union of functions on disjoint domains. (Contributed by Thierry Arnoux, 30-Mar-2017.)
(𝑥𝐴 ↦ if(𝑥𝐵, 𝐶, 𝐷)) = ((𝑥 ∈ (𝐴𝐵) ↦ 𝐶) ∪ (𝑥 ∈ (𝐴𝐵) ↦ 𝐷))

Theoremdfcnv2 28859* Alternative definition of the converse of a relation. (Contributed by Thierry Arnoux, 31-Mar-2018.)
(ran 𝑅𝐴𝑅 = 𝑥𝐴 ({𝑥} × (𝑅 “ {𝑥})))

Theoremmpt2mptxf 28860* Express a two-argument function as a one-argument function, or vice-versa. In this version 𝐵(𝑥) is not assumed to be constant w.r.t 𝑥. (Contributed by Mario Carneiro, 29-Dec-2014.) (Revised by Thierry Arnoux, 31-Mar-2018.)
𝑥𝐶    &   𝑦𝐶    &   (𝑧 = ⟨𝑥, 𝑦⟩ → 𝐶 = 𝐷)       (𝑧 𝑥𝐴 ({𝑥} × 𝐵) ↦ 𝐶) = (𝑥𝐴, 𝑦𝐵𝐷)

Theoremgtiso 28861 Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.)
((𝐴 ⊆ ℝ*𝐵 ⊆ ℝ*) → (𝐹 Isom < , < (𝐴, 𝐵) ↔ 𝐹 Isom ≤ , ≤ (𝐴, 𝐵)))

Theoremisoun 28862* Infer an isomorphism from a union of two isomorphisms. (Contributed by Thierry Arnoux, 30-Mar-2017.)
(𝜑𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵))    &   (𝜑𝐺 Isom 𝑅, 𝑆 (𝐶, 𝐷))    &   ((𝜑𝑥𝐴𝑦𝐶) → 𝑥𝑅𝑦)    &   ((𝜑𝑧𝐵𝑤𝐷) → 𝑧𝑆𝑤)    &   ((𝜑𝑥𝐶𝑦𝐴) → ¬ 𝑥𝑅𝑦)    &   ((𝜑𝑧𝐷𝑤𝐵) → ¬ 𝑧𝑆𝑤)    &   (𝜑 → (𝐴𝐶) = ∅)    &   (𝜑 → (𝐵𝐷) = ∅)       (𝜑 → (𝐻𝐺) Isom 𝑅, 𝑆 ((𝐴𝐶), (𝐵𝐷)))

21.3.4.5  Disjointness (additional proof requiring functions)

Theoremdisjdsct 28863* A disjoint collection is distinct, i.e. each set in this collection is different of all others, provided that it does not contain the empty set This can be expressed as "the converse of the mapping function is a function", or "the mapping function is single-rooted". (Cf. funcnv 5872) (Contributed by Thierry Arnoux, 28-Feb-2017.)
𝑥𝜑    &   𝑥𝐴    &   ((𝜑𝑥𝐴) → 𝐵 ∈ (𝑉 ∖ {∅}))    &   (𝜑Disj 𝑥𝐴 𝐵)       (𝜑 → Fun (𝑥𝐴𝐵))

21.3.4.6  First and second members of an ordered pair - misc additions

Theoremdf1stres 28864* Definition for a restriction of the 1st (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
(1st ↾ (𝐴 × 𝐵)) = (𝑥𝐴, 𝑦𝐵𝑥)

Theoremdf2ndres 28865* Definition for a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
(2nd ↾ (𝐴 × 𝐵)) = (𝑥𝐴, 𝑦𝐵𝑦)

Theorem1stpreimas 28866 The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.)
((Rel 𝐴𝑋𝑉) → ((1st𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))

Theorem1stpreima 28867 The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
(𝐴𝐵 → ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))

Theorem2ndpreima 28868 The preimage by 2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
(𝐴𝐶 → ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))

Theoremcurry2ima 28869* The image of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 25-Sep-2017.)
𝐺 = (𝐹(1st ↾ (V × {𝐶})))       ((𝐹 Fn (𝐴 × 𝐵) ∧ 𝐶𝐵𝐷𝐴) → (𝐺𝐷) = {𝑦 ∣ ∃𝑥𝐷 𝑦 = (𝑥𝐹𝐶)})

Theoremsupssd 28870* Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐵𝐶)    &   (𝜑𝐶𝐴)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐵 𝑦𝑅𝑧)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦𝐴 (𝑦𝑅𝑥 → ∃𝑧𝐶 𝑦𝑅𝑧)))       (𝜑 → ¬ sup(𝐶, 𝐴, 𝑅)𝑅sup(𝐵, 𝐴, 𝑅))

Theoreminfssd 28871* Inequality deduction for infimum of a subset. (Contributed by AV, 4-Oct-2020.)
(𝜑𝑅 Or 𝐴)    &   (𝜑𝐶𝐵)    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐶 𝑧𝑅𝑦)))    &   (𝜑 → ∃𝑥𝐴 (∀𝑦𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦𝐴 (𝑥𝑅𝑦 → ∃𝑧𝐵 𝑧𝑅𝑦)))       (𝜑 → ¬ inf(𝐶, 𝐴, 𝑅)𝑅inf(𝐵, 𝐴, 𝑅))

21.3.4.8  Finite Sets

Theoremimafi2 28872 The image by a finite set is finite. See also imafi 8142. (Contributed by Thierry Arnoux, 25-Apr-2020.)
(𝐴 ∈ Fin → (𝐴𝐵) ∈ Fin)

Theoremunifi3 28873 If a union is finite, then all its elements are finite. See unifi 8138. (Contributed by Thierry Arnoux, 27-Aug-2017.)
( 𝐴 ∈ Fin → 𝐴 ⊆ Fin)

21.3.4.9  Countable Sets

Theoremsnct 28874 A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
(𝐴𝑉 → {𝐴} ≼ ω)

Theoremprct 28875 An unordered pair is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.)
((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} ≼ ω)

Theoremfnct 28876 If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
((𝐹 Fn 𝐴𝐴 ≼ ω) → 𝐹 ≼ ω)

Theoremdmct 28877 The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → dom 𝐴 ≼ ω)

Theoremcnvct 28878 If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → 𝐴 ≼ ω)

Theoremrnct 28879 The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → ran 𝐴 ≼ ω)

Theoremmptct 28880* A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → (𝑥𝐴𝐵) ≼ ω)

Theoremmpt2cti 28881* An operation is countable if both its domains are countable. (Contributed by Thierry Arnoux, 17-Sep-2017.)
𝑥𝐴𝑦𝐵 𝐶𝑉       ((𝐴 ≼ ω ∧ 𝐵 ≼ ω) → (𝑥𝐴, 𝑦𝐵𝐶) ≼ ω)

Theoremabrexct 28882* An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
(𝐴 ≼ ω → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≼ ω)

Theoremmptctf 28883 A countable mapping set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝐴       (𝐴 ≼ ω → (𝑥𝐴𝐵) ≼ ω)

Theoremabrexctf 28884* An image set of a countable set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
𝑥𝐴       (𝐴 ≼ ω → {𝑦 ∣ ∃𝑥𝐴 𝑦 = 𝐵} ≼ ω)

Theorempadct 28885* Index a countable set with integers and pad with 𝑍. (Contributed by Thierry Arnoux, 1-Jun-2020.)
((𝐴 ≼ ω ∧ 𝑍𝑉 ∧ ¬ 𝑍𝐴) → ∃𝑓(𝑓:ℕ⟶(𝐴 ∪ {𝑍}) ∧ 𝐴 ⊆ ran 𝑓 ∧ Fun (𝑓𝐴)))

Theoremcnvoprab 28886* The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)
𝑥𝜓    &   𝑦𝜓    &   (𝑎 = ⟨𝑥, 𝑦⟩ → (𝜓𝜑))    &   (𝜓𝑎 ∈ (V × V))       {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} = {⟨𝑧, 𝑎⟩ ∣ 𝜓}

Theoremf1od2 28887* Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)
𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)    &   ((𝜑 ∧ (𝑥𝐴𝑦𝐵)) → 𝐶𝑊)    &   ((𝜑𝑧𝐷) → (𝐼𝑋𝐽𝑌))    &   (𝜑 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ (𝑧𝐷 ∧ (𝑥 = 𝐼𝑦 = 𝐽))))       (𝜑𝐹:(𝐴 × 𝐵)–1-1-onto𝐷)

Theoremfcobij 28888* Composing functions with a bijection yields a bijection between sets of functions. (Contributed by Thierry Arnoux, 25-Aug-2017.)
(𝜑𝐺:𝑆1-1-onto𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑆𝑉)    &   (𝜑𝑇𝑊)       (𝜑 → (𝑓 ∈ (𝑆𝑚 𝑅) ↦ (𝐺𝑓)):(𝑆𝑚 𝑅)–1-1-onto→(𝑇𝑚 𝑅))

Theoremfcobijfs 28889* Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 8196. (Contributed by Thierry Arnoux, 25-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
(𝜑𝐺:𝑆1-1-onto𝑇)    &   (𝜑𝑅𝑈)    &   (𝜑𝑆𝑉)    &   (𝜑𝑇𝑊)    &   (𝜑𝑂𝑆)    &   𝑄 = (𝐺𝑂)    &   𝑋 = {𝑔 ∈ (𝑆𝑚 𝑅) ∣ 𝑔 finSupp 𝑂}    &   𝑌 = { ∈ (𝑇𝑚 𝑅) ∣ finSupp 𝑄}       (𝜑 → (𝑓𝑋 ↦ (𝐺𝑓)):𝑋1-1-onto𝑌)

Theoremsuppss3 28890* Deduce a function's support's inclusion in another function's support. (Contributed by Thierry Arnoux, 7-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐺 = (𝑥𝐴𝐵)    &   (𝜑𝐴𝑉)    &   (𝜑𝑍𝑊)    &   (𝜑𝐹 Fn 𝐴)    &   ((𝜑𝑥𝐴 ∧ (𝐹𝑥) = 𝑍) → 𝐵 = 𝑍)       (𝜑 → (𝐺 supp 𝑍) ⊆ (𝐹 supp 𝑍))

Theoremffs2 28891 Rewrite a function's support based with its range rather than the universal class. See also frnsuppeq 7194. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐶 = (𝐵 ∖ {𝑍})       ((𝐴𝑉𝑍𝑊𝐹:𝐴𝐵) → (𝐹 supp 𝑍) = (𝐹𝐶))

Theoremffsrn 28892 The range of a finitely supported function is finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)
(𝜑𝑍𝑊)    &   (𝜑𝐹𝑉)    &   (𝜑 → Fun 𝐹)    &   (𝜑 → (𝐹 supp 𝑍) ∈ Fin)       (𝜑 → ran 𝐹 ∈ Fin)

Theoremresf1o 28893* Restriction of functions to a superset of their support creates a bijection. (Contributed by Thierry Arnoux, 12-Sep-2017.)
𝑋 = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ (𝑓 “ (𝐵 ∖ {𝑍})) ⊆ 𝐶}    &   𝐹 = (𝑓𝑋 ↦ (𝑓𝐶))       (((𝐴𝑉𝐵𝑊𝐶𝐴) ∧ 𝑍𝐵) → 𝐹:𝑋1-1-onto→(𝐵𝑚 𝐶))

Theoremmaprnin 28894* Restricting the range of the mapping operator. (Contributed by Thierry Arnoux, 30-Aug-2017.)
𝐴 ∈ V    &   𝐵 ∈ V       ((𝐵𝐶) ↑𝑚 𝐴) = {𝑓 ∈ (𝐵𝑚 𝐴) ∣ ran 𝑓𝐶}

Theoremfpwrelmapffslem 28895* Lemma for fpwrelmapffs 28897. For this theorem, the sets 𝐴 and 𝐵 could be infinite, but the relation 𝑅 itself is finite. (Contributed by Thierry Arnoux, 1-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐴 ∈ V    &   𝐵 ∈ V    &   (𝜑𝐹:𝐴⟶𝒫 𝐵)    &   (𝜑𝑅 = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐹𝑥))})       (𝜑 → (𝑅 ∈ Fin ↔ (ran 𝐹 ⊆ Fin ∧ (𝐹 supp ∅) ∈ Fin)))

Theoremfpwrelmap 28896* Define a canonical mapping between functions from 𝐴 into subsets of 𝐵 and the relations with domain 𝐴 and range within 𝐵. Note that the same relation is used in axdc2lem 9153 and marypha2lem1 8224. (Contributed by Thierry Arnoux, 28-Aug-2017.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑀 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})       𝑀:(𝒫 𝐵𝑚 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)

Theoremfpwrelmapffs 28897* Define a canonical mapping between finite relations (finite subsets of a cartesian product) and functions with finite support into finite subsets. (Contributed by Thierry Arnoux, 28-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
𝐴 ∈ V    &   𝐵 ∈ V    &   𝑀 = (𝑓 ∈ (𝒫 𝐵𝑚 𝐴) ↦ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝑓𝑥))})    &   𝑆 = {𝑓 ∈ ((𝒫 𝐵 ∩ Fin) ↑𝑚 𝐴) ∣ (𝑓 supp ∅) ∈ Fin}       (𝑀𝑆):𝑆1-1-onto→(𝒫 (𝐴 × 𝐵) ∩ Fin)

21.3.5  Real and Complex Numbers

21.3.5.1  Complex operations - misc. additions

Theoremaddeq0 28898 Two complex which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 2-May-2017.)
((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) = 0 ↔ 𝐴 = -𝐵))

Theoremsubeqxfrd 28899 Transfer two terms of a subtraction in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝜑𝐴 ∈ ℂ)    &   (𝜑𝐵 ∈ ℂ)    &   (𝜑𝐶 ∈ ℂ)    &   (𝜑𝐷 ∈ ℂ)    &   (𝜑 → (𝐴𝐵) = (𝐶𝐷))       (𝜑 → (𝐴𝐶) = (𝐵𝐷))

Theoremznsqcld 28900 Squaring of nonzero relative numbers. (Contributed by Thierry Arnoux, 2-Feb-2020.)
(𝜑𝑁 ∈ ℤ)    &   (𝜑𝑁 ≠ 0)       (𝜑 → (𝑁↑2) ∈ ℕ)

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